Annali di Matematica Pura ed Applicata

, Volume 186, Issue 1, pp 85–98 | Cite as

Asymptotic behaviour in periodic three species predator–prey systems

Article

Abstract

For a three-dimensional periodic Lotka–Volterra system, the asymptotic behaviour of its positive solutions is investigated. More specifically, we give suitable average conditions which lead to extinction of the competitively inferior species and global stable coexistence of the two remaining predator and prey species.

Keywords

Predator–prey model Three species community Periodic systems Extinction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahmad, S., Lazer, A.C.: Average conditions for global asymptotic stability in a nonautonomous Lotka—Volterra system. Nonlinear Anal. 40, 37–49 (2000)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cushing, J.M.: Periodic time-dependent predator—prey systems. SIAM J. Appl. Math. 32, 82–95 (1977)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Korman, P.: On a principle of predatory exclusion. Appl. Anal. 84, 707–712 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Korobeinikov, A., Wake, G.C.: Global properties of the three-dimensional predator—prey Lotka—Volterra systems. J. Appl. Math. Decis. Sci. 3, 155–162 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Krikorian, N.: The Volterra model for three species predator—prey systems:boundedness and stability. J. Math. Biol. 7, 117–132 (1979)MATHMathSciNetGoogle Scholar
  6. 6.
    Lisena, B.: Global stability in competitive periodic systems. Nonlinear Anal. RWA 5, 613–627 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lisena, B.: Global attractive periodic models of predator—prey type. Nonlinear Anal. 6, 133–144 (2005)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lisena, B.: Extinction in three species competitive systems with periodic coefficients. Dynamic Syst. 14, 393–406 (2005)MATHMathSciNetGoogle Scholar
  9. 9.
    Takuchi, Y., Adachi, N.: Existence and bifurcation of stable equilibrium in two-prey, one-predator communities. Bull. Math. Biol. 45, 877–900 (1983)MathSciNetGoogle Scholar
  10. 10.
    Yang, P., Xu, R.: Global attractivity of the periodic Lotka—Volterra system. J. Math. Anal. Appl. 233, 221–232 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Zhao, J., Jiang, J.: Permanence in nonautonomous Lotka—Volterra system with predator—prey. Appl. Math. Comput. 152, 99–109 (2004)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversitá degli studi di BariBariItaly

Personalised recommendations